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Theorem pnt 24315
Description: The Prime Number Theorem: the number of prime numbers less than  x tends asymptotically to  x  /  log (
x ) as  x goes to infinity. This is Metamath 100 proof #5. (Contributed by Mario Carneiro, 1-Jun-2016.)
Assertion
Ref Expression
pnt  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1

Proof of Theorem pnt
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 9641 . . . . . . 7  |-  1  e.  RR
21rexri 9692 . . . . . 6  |-  1  e.  RR*
3 1lt2 10776 . . . . . 6  |-  1  <  2
4 df-ioo 11639 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
5 df-ico 11641 . . . . . . 7  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
6 xrltletr 11454 . . . . . . 7  |-  ( ( 1  e.  RR*  /\  2  e.  RR*  /\  w  e. 
RR* )  ->  (
( 1  <  2  /\  2  <_  w )  ->  1  <  w
) )
74, 5, 6ixxss1 11653 . . . . . 6  |-  ( ( 1  e.  RR*  /\  1  <  2 )  ->  (
2 [,) +oo )  C_  ( 1 (,) +oo ) )
82, 3, 7mp2an 676 . . . . 5  |-  ( 2 [,) +oo )  C_  ( 1 (,) +oo )
9 resmpt 5174 . . . . 5  |-  ( ( 2 [,) +oo )  C_  ( 1 (,) +oo )  ->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) ) )
108, 9mp1i 13 . . . 4  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) )
118sseli 3466 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  ( 1 (,) +oo ) )
12 ioossre 11696 . . . . . . . . . . 11  |-  ( 1 (,) +oo )  C_  RR
1312sseli 3466 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR )
1411, 13syl 17 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR )
15 2re 10679 . . . . . . . . . . 11  |-  2  e.  RR
16 pnfxr 11412 . . . . . . . . . . 11  |- +oo  e.  RR*
17 elico2 11698 . . . . . . . . . . 11  |-  ( ( 2  e.  RR  /\ +oo  e.  RR* )  ->  (
x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) ) )
1815, 16, 17mp2an 676 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  <->  ( x  e.  RR  /\  2  <_  x  /\  x  < +oo ) )
1918simp2bi 1021 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  2  <_  x )
20 chtrpcl 23965 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
( theta `  x )  e.  RR+ )
2114, 19, 20syl2anc 665 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  RR+ )
22 0red 9643 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  e.  RR )
231a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  e.  RR )
24 0lt1 10135 . . . . . . . . . . . 12  |-  0  <  1
2524a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  0  <  1 )
26 eliooord 11694 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( 1  <  x  /\  x  < +oo ) )
2726simpld 460 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  1  < 
x )
2822, 23, 13, 25, 27lttrd 9795 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  0  < 
x )
2913, 28elrpd 11338 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR+ )
3011, 29syl 17 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  RR+ )
3121, 30rpdivcld 11358 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  /  x )  e.  RR+ )
3231adantl 467 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  x )  e.  RR+ )
33 ppinncl 23964 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  2  <_  x )  -> 
(π `  x )  e.  NN )
3414, 19, 33syl2anc 665 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  NN )
3534nnrpd 11339 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  RR+ )
3613, 27rplogcld 23443 . . . . . . . . . 10  |-  ( x  e.  ( 1 (,) +oo )  ->  ( log `  x )  e.  RR+ )
3711, 36syl 17 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  RR+ )
3835, 37rpmulcld 11357 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  RR+ )
3921, 38rpdivcld 11358 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  e.  RR+ )
4039adantl 467 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  e.  RR+ )
4130ssriv 3474 . . . . . . . 8  |-  ( 2 [,) +oo )  C_  RR+
42 resmpt 5174 . . . . . . . 8  |-  ( ( 2 [,) +oo )  C_  RR+  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) ) )
4341, 42ax-mp 5 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x )  /  x ) )
44 pnt2 24314 . . . . . . . 8  |-  ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1
45 rlimres 13600 . . . . . . . 8  |-  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  ~~> r  1  ->  ( ( x  e.  RR+  |->  ( (
theta `  x )  /  x ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
4644, 45mp1i 13 . . . . . . 7  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( ( theta `  x )  /  x
) )  |`  (
2 [,) +oo )
)  ~~> r  1 )
4743, 46syl5eqbrr 4460 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  x ) )  ~~> r  1 )
48 chtppilim 24176 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  |->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) )  ~~> r  1
4948a1i 11 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( theta `  x
)  /  ( (π `  x )  x.  ( log `  x ) ) ) )  ~~> r  1 )
50 ax-1ne0 9607 . . . . . . 7  |-  1  =/=  0
5150a1i 11 . . . . . 6  |-  ( T. 
->  1  =/=  0
)
5239rpne0d 11346 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) )  =/=  0 )
5352adantl 467 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 2 [,) +oo ) )  ->  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) )  =/=  0 )
5432, 40, 47, 49, 51, 53rlimdiv 13687 . . . . 5  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x
)  /  ( (
theta `  x )  / 
( (π `  x )  x.  ( log `  x
) ) ) ) )  ~~> r  ( 1  /  1 ) )
5514recnd 9668 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  x  e.  CC )
56 chtcl 23899 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  ( theta `  x )  e.  RR )
5713, 56syl 17 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  RR )
5857recnd 9668 . . . . . . . . . . 11  |-  ( x  e.  ( 1 (,) +oo )  ->  ( theta `  x )  e.  CC )
5911, 58syl 17 . . . . . . . . . 10  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  e.  CC )
6055, 59mulcomd 9663 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( x  x.  ( theta `  x
) )  =  ( ( theta `  x )  x.  x ) )
6160oveq2d 6321 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) ) )
6238rpcnd 11343 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  e.  CC )
6330rpne0d 11346 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  x  =/=  0 )
6421rpne0d 11346 . . . . . . . . 9  |-  ( x  e.  ( 2 [,) +oo )  ->  ( theta `  x )  =/=  0
)
6562, 55, 59, 63, 64divcan5d 10408 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( ( theta `  x
)  x.  x ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6661, 65eqtrd 2470 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x
) )  /  x
) )
6738rpne0d 11346 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  x.  ( log `  x ) )  =/=  0 )
6859, 55, 59, 62, 63, 67, 64divdivdivd 10429 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( ( ( theta `  x )  x.  ( (π `  x )  x.  ( log `  x
) ) )  / 
( x  x.  ( theta `  x ) ) ) )
6934nncnd 10625 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  (π `  x
)  e.  CC )
7037rpcnd 11343 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  e.  CC )
7137rpne0d 11346 . . . . . . . 8  |-  ( x  e.  ( 2 [,) +oo )  ->  ( log `  x )  =/=  0
)
7269, 55, 70, 63, 71divdiv2d 10414 . . . . . . 7  |-  ( x  e.  ( 2 [,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  =  ( ( (π `  x )  x.  ( log `  x ) )  /  x ) )
7366, 68, 723eqtr4d 2480 . . . . . 6  |-  ( x  e.  ( 2 [,) +oo )  ->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) )  =  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
7473mpteq2ia 4508 . . . . 5  |-  ( x  e.  ( 2 [,) +oo )  |->  ( ( ( theta `  x )  /  x )  /  (
( theta `  x )  /  ( (π `  x
)  x.  ( log `  x ) ) ) ) )  =  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )
75 1div1e1 10299 . . . . 5  |-  ( 1  /  1 )  =  1
7654, 74, 753brtr3g 4457 . . . 4  |-  ( T. 
->  ( x  e.  ( 2 [,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
7710, 76eqbrtrd 4446 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  |`  ( 2 [,) +oo ) )  ~~> r  1 )
78 ppicl 23921 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (π `  x )  e.  NN0 )
7913, 78syl 17 . . . . . . . . 9  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  NN0 )
8079nn0red 10926 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  (π `  x
)  e.  RR )
8129, 36rpdivcld 11358 . . . . . . . 8  |-  ( x  e.  ( 1 (,) +oo )  ->  ( x  /  ( log `  x
) )  e.  RR+ )
8280, 81rerpdivcld 11369 . . . . . . 7  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  RR )
8382recnd 9668 . . . . . 6  |-  ( x  e.  ( 1 (,) +oo )  ->  ( (π `  x )  /  (
x  /  ( log `  x ) ) )  e.  CC )
8483adantl 467 . . . . 5  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
(π `  x )  / 
( x  /  ( log `  x ) ) )  e.  CC )
85 eqid 2429 . . . . 5  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )
8684, 85fmptd 6061 . . . 4  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) ) : ( 1 (,) +oo ) --> CC )
8712a1i 11 . . . 4  |-  ( T. 
->  ( 1 (,) +oo )  C_  RR )
8815a1i 11 . . . 4  |-  ( T. 
->  2  e.  RR )
8986, 87, 88rlimresb 13607 . . 3  |-  ( T. 
->  ( ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x
)  /  ( x  /  ( log `  x
) ) ) )  ~~> r  1  <->  ( (
x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  |`  (
2 [,) +oo )
)  ~~> r  1 ) )
9077, 89mpbird 235 . 2  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  / 
( x  /  ( log `  x ) ) ) )  ~~> r  1 )
9190trud 1446 1  |-  ( x  e.  ( 1 (,) +oo )  |->  ( (π `  x )  /  (
x  /  ( log `  x ) ) ) )  ~~> r  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ w3a 982    = wceq 1437   T. wtru 1438    e. wcel 1870    =/= wne 2625    C_ wss 3442   class class class wbr 4426    |-> cmpt 4484    |` cres 4856   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543   +oocpnf 9671   RR*cxr 9673    < clt 9674    <_ cle 9675    / cdiv 10268   NNcn 10609   2c2 10659   NN0cn0 10869   RR+crp 11302   (,)cioo 11635   [,)cico 11637    ~~> r crli 13527   logclog 23369   thetaccht 23880  πcppi 23883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-o1 13532  df-lo1 13533  df-sum 13731  df-ef 14099  df-e 14100  df-sin 14101  df-cos 14102  df-pi 14104  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-cmp 20333  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699  df-log 23371  df-cxp 23372  df-em 23783  df-cht 23886  df-vma 23887  df-chp 23888  df-ppi 23889  df-mu 23890
This theorem is referenced by: (None)
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